The isotropic XYXY model on the inhomogeneous periodic chain

The static and dynamic properties of the isotropic XYXY-model (s=12) on the inhomogeneous periodic chain, composed of NN segments with nn different exchange interactions and magnetic moments, in a transverse field hh, are obtained exactly at arbitrary temperatures. The properties are determined by introducing the generalized Jordan–Wigner transformation and by reducing the problem to a diagonalization of a finite matrix of nnth order. The diagonalization procedure is discussed in detail and the critical behaviour induced by the transverse field, at T=0T=0, is presented. The quantum transitions are determined by analyzing the behaviour of the induced magnetization, defined as (1/n)∑m=1nμm〈Sj,mz〉 where μmμm is the magnetic moment at site mm within the segment jj, as a function of the field, and the critical fields determined exactly. The dynamic correlations, 〈Sj,mz(t)Sj′,m′z(0)〉, and the dynamic susceptibility χqzz(ω) are also obtained at arbitrary temperatures. Explicit results are presented in the limit T=0T=0, where the critical behaviour occurs, for the static susceptibility χqzz(0) as a function of the transverse field hh, and for the frequency dependency of dynamic susceptibility χqzz(ω). Also in this limit, the transverse time-correlation 〈Sj,mx(t)Sj′,m′x(0)〉 and the dynamic and isothermal susceptibilities, χqxx(ω) and χTxx, are obtained for the transverse field greater or equal than the saturation field.